Coloring linear hypergraphs: the Erdős–Faber–Lovász conjecture and the Combinatorial Nullstellensatz

Designs, Codes and Cryptography - The long-standing Erdős–Faber–Lovász conjecture states that every n-uniform linear hypergaph with n edges has a proper vertex-coloring using...     

Orientations of chain groups

We prove generalizations to chain groups, of Minty's Arc Colouring Lemma and its extension, the well-known Farkas Lemma. In these the orientation of the edges is replaced by an arbitrary chain.A function on a chain groupN isrepresentable if there exists a chainR such that (X)=R·X for allXN. Anorientation is a chain with values ±1. We prove that for a regular chain group a linear function that is representable by an orientation for each chainXN locally, is representable by an orientation globally.     

(Z)-1-[2-(Triarylstannyl)vinyl]-1-cycloheptanols: Synthesis, characterization, halodemetallation and crystal structures

The synthesis and characterization by ¹H, ¹³C, ¹¹⁹Sn NMR and ¹¹⁹Sn Mössbauer spectroscopy of (Z)-1-[2-(triphenylstannyl)vinyl]-1-cycloheptanol,

Full-size image

(1), and (Z)-1-[2-tri-p-tolylstannyl)vinyl-1-cycloheptanol,

Full-size image

(2), are described, together with their halodemetallation by I₂, Br₂ and ICIl to yield derivatives of the types

Full-size image

(Ar = phenyl or p-tolyl, N = 1, 2; X = I, Br, Cl, respectively). The solid-state structures of four compounds have been determined by X-ray diffraction analysis. In the crystals of

Full-size image

(1) and

Full-size image

(2) the Sn atom has a tetrahedral geometry distorted towards trigonal bipyramid as a consequence of a close intramolecular contact with the hydroxyl O(1) atom of 2.742(3) Å and 2.768(3) Å, respectively. A trigonal bipyramidal geometry is found in

Full-size image

(12) and

Full-size image

(4), in which significant Sn---O(1) interactions are noted [2.437(8) Å and 2.407(8) Å, respectively].     

Tensor products of contexts and complete lattices

Although the categoryCLC of complete lattices and complete homomorphisms does not possess arbitrary coproducts, we show that the tensor product introduced by Wille has the universal property of coproducts for so-called distributing families of morphisms (and only for these). As every family of morphisms into a completely distributive lattice is distributing, this includes the known fact that in the category of completely distributive lattices, arbitrary coproducts exist and coincide with the tensor products. Since the definition of tensor products is based on the notion of contexts and their concept lattices, many results on tensor products extend from complete lattices to contexts. Thus we introduce two kinds of tensor products for arbitrary families of contexts, a partial and a complete one, and establish universal properties of these tensor products.Presented by B. Jonsson.     

Modelling of the processes of ultrafine SiO2 production under thermal plasma conditions

The production of ultrafine silica particles is examined by modelling the processes in a flow plasma reactor. The model is based on the authors' experimental data on silica sand processing by use of thermal arc plasma. The free-molecular coagulation is assumed to be the dominant process for particle growth. This is carried out at fast cooling of the vapour, during its mixing with oxygen. The particle size distribution functions are calculated, and the influence of the chemical monomer generation and the mixing on their behavior is investigated. Comparison of the calculated and the experimental mean particle size is made.List of symbols C _nl collision frequency function, cm³/s - d _m mean median diameter, nm - F _LN(d) integral form of the particle size distribution function,% - g(t) mixing function - k Boltzmann's constant - k ₁,k,k rate coefficients, cm³/s - k ₂,k ₃ rate coefficiens, cm⁶/s - m mass of the monomer, g - M number of groups - T absolute temperature, K - T ₀ temperature of the hot vapour, K - T _E temperature of SiO₂ condensation, K - t time, s - t _m mixing time, s - t _c colling time, s - z ₀ monomer concentration, cm^–3 - z _n ,z _l concentrations of particles in groups, cm^–3 - z _n total number concentration of particles (n=0, 1, M), cm^–3 - z _n /(z _n )(log d _n+1 –logd _n) differential form of particle size distribution function - w _i reaction rates (i=1, 2, 3), cm^–3/s - , cooling rates, K/s - coefficient of proportionality - _SiO conversion rate of SiO,% - coefficient determined from the cooling rate, s^–1 - mass density of the particles, g/cm³ - standard deviation The authors gratefully acknowledge Prof. L. S. Polak from the Institute of Petrol Chemical Synthesis, Russian Academy of Science, for helpful discussions.     

Hyperfine interactions in icosahedral quasicrystals: a Mössbauer study

Icosahedral Al₆₅Cu₂₀Fe₁₅ and Al₄₀Cu_9.9Ge₂₅Mn₂₅ ⁵⁷Fe_0.1 quasicrystals are studied using⁵⁷Fe transmission Mössbauer experiments. The spectra are analyzed by distributions of electric-quadrupole interaction accounting for line asymmetries. Temperature dependences of the hyperfine parameters derived comprising average values ofP() distributions, corresponding standard deviations and center shifts are presented in a whole range from 8 to 300 K.     

Local order of icosahedral quasicrystals studied by Mössbauer spectroscopy

Thermodynamically stable Al-Cu-Fe and Fe-doped ferromagnetic Al-Cu-Ge-Mn icosahedral quasicrystals are studied by⁵⁷Fe transmission Mössbauer spectroscopy and X-ray diffraction experiments. Al₆₅Cu₂₀Fe₁₅ quasicrystalline alloy was subjected to a mechanical grinding (MG) for up to 800 hours in a ball mill. Presence of the amorphous phase which co-exists with the quasicrystalline one is revealed in the early stage of MG. Mössbauer measurements were performed on icosahedral Al₄₀Cu_10–x Ge₂₅Mn₂₅Fe _x quasicrystal (x0.1; 3) in a temperature range from 10 K to 548 K. It was found that a magnetic transition occurs at about 30 K which is far belowT _c reported in the literature. It is concluded that AlGeMn ferromagnet which is present in the samples does not affect the magnetic transition observed and the transition is an intrinsic property of the Al-Cu-Ge-Mn host alloy.Samples of icosahedral quasicrystals were kindly provided by Profs. A. Inoue, T. Masumoto and P. H. Shingu. Ball milling was performed in Kyoto University by a courtesy of Prof. P. H. Shingu. This work was supported by the project for priority areas on properties of quasicrystals (No. 01630003) from the Japanese Ministry of Education, Science and Culture.     

Sur de nouvelles phases séléniées ternaires du molybdène

The synthesis and the crystal properties of new selenides of formula M_xMo₃Se₄ are described. If M = Zn, Ag, Cd, Sn and Pb, they are stoichiometric with x = 0.6; if M = Fe, Mn, Cr, V, Ti, triclinic solid solutions are observed with 0.5 < x < 0.7; if M = Cu, Co, Ni, rhombohedral solid solutions are obtained with 0 < x < 1.4 for M = Cu, 0 < x < 0.7 for M = Co and 0 < x < 0.8 for M = Ni All these phases can be deduced from the Mo₃Se₄ structure by introducing metal atoms into the tunnels between the “Mo₆Se₈” metal atom cluster configuration.